Heat conduction: a telegraph-type model with self-similar behavior of solutions

TL;DR

This paper introduces a modified heat conduction model based on a non-autonomous telegraph-type equation, which admits self-similar solutions that could serve as intermediate asymptotics, expanding understanding of heat flux dynamics.

Contribution

It proposes a novel non-autonomous telegraph-type heat conduction model with self-similar solutions, extending classical Fourier and Cattaneo laws.

Findings

Derivation of a non-autonomous telegraph-type equation for heat conduction.

Identification of self-similar solutions as products of traveling waves.

Potential role of these solutions as intermediate asymptotics.

Abstract

For heat flux $q$ and temperature $T$ we introduce a modified Fourier--Cattaneo law $q_t+ l \frac{q}{t}= - kT_x .$ The consequence of it is a non-autonomous telegraph-type equation. % $\epsilon S_{tt} + \frac{a}{t} S_t = S_{xx}$ . This model already has a typical self-similar solution which may be written as product of two travelling waves modulo a time-dependent factor and might play a role of intermediate asymptotics.